Three-dimensional reachable set for the Dubins car: Foundation of analytical description
Communications in Optimization Theory · 2022
The Dubins car is a model of motion, in which the scalar control u determines the instantaneous angular speed of rotation.The paper considers the symmetric variant of the constraints u∈[u 1 ,u 2 ], where u 1 = -u 2 and u 2 = 1.We study the three-dimensional reachable set at a given instant t f > 0. An analytical description of twodimensional sections of the set w.r.t. the angular coordinate ϕ is given.The boundary of each ϕ-section is formed with the help of a certain set of curves obtained using the Pontryagin maximum principle.This set includes circular arcs and circular involutes.The symmetry property of each ϕ-section w.r.t. a certain straight line is established.Classification of possible types of the ϕ-sections is proposed.The greatest difficulty is presented by analysis of the case with non-simply connected ϕ-sections.The range of values ϕ and t f , at which the ϕ-sections are non-simply connected, is indicated.Keywords.Dubins car